Abstract
A modified modulated wideband converter (MWC)based compressed sampling receiver and a parameter estimation algorithm in discrete time fractional Fourier domain (DTFrFD) to intercept and estimate the fractional bandlimited LFM signals are suggested in this work. Our proposed compressed sampling receiver has some advantages compared to the original MWCbased compressed sampling receiver. Firstly, the proposed receiver works in DTFrFD which is more effective to intercept the fractional bandlimited signals, while the original MWCbased digital receiver works in discrete time Fourier domain (DTFD). Secondly, the crosschannel signal can get better separation by the proposed digital receiver than by original MWCbased receiver because that the nonzero part of the signal’s available spectrum is narrower in DTFrFD than in DTFD. Then, with the data acquired from the digital receiver, we propose a novel parameter estimation algorithm for the intercepted fractional bandlimited LFM signal based on OMP and the differentiation spectrum in DTFrFD. The initial frequency and final frequency of the intercepted fractional bandlimited LFM signal can be extracted successfully without twodimensional search and reconstruction, and only two iterations are needed according the algorithm. Both theoretical analysis and simulation results demonstrate that the proposed algorithm bears a relatively low complexity and its estimation precision is not only higher than both DFrFTbased search algorithm and DFrFTbased reconstruction algorithm but also close to Cramer–Rao lower bound.
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Abbreviations
 MWC:

Modulated wideband converter
 DFrFT:

Digital fractional Fourier transform
 LFM:

Linear frequency modulation
 DTFrFD:

Discrete time fractional Fourier domain
 OMP:

Orthogonal matching pursuit
 DTFrFT:

Discrete time fractional Fourier transform
 DTFT:

Discrete time Fourier transform
 DTFD:

Discrete time Fourier domain
 MSE:

Mean squared error
 LPI:

Low probability of intercept
 SNR:

Signaltonoise ratio
 FrFT:

Fractional Fourier transform
 FrFD:

Fractional Fourier domain
 SFrFT:

Simplified fractional Fourier transform
 FT:

Fourier transform
 DSFrFT:

Digital simplified fractional Fourier transform
 SFrFD:

Simplified fractional Fourier domain
 AWGN:

Additive white Gaussian noise
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Acknowledgements
This research was funded by National Natural Science Foundation of China Grant No. 61271354 and Henan Province Science and Technology Key Project Grant No. 142102210431.
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Appendix A
Appendix A
Spectrum distribution characteristics of LFM signal in SFrFD A monocomponent LFM signal is defined by:
where \(f_0\) is the initial frequency. A is the amplitude of \(x\left( t\right) \) which could berandom or fixed. \(K_{\mathrm{lfm}}\) is the modulation rate. And the duration time of \(x\left( t\right) \) is \( \left[ \frac{T_d}{2},\frac{T_d}{2}\right] \).
The SFrFT of \(x\left( t\right) \) can be calculated by
According to (S2), \({\overline{X}}_\alpha \left( u\right) \), the SFrFT of \( x\left( u\right) \), denotes the FT of another LFM signal whose modulation rate is \( K'_{\mathrm{lfm}} =K_{\mathrm{lfm}}+\cot \alpha /2 \pi \), initial frequency is \(f_0\), and the frequency interval is \(\left[ f_0K'_{\mathrm{lfm}}\frac{T_d}{2}, f_0+K'_{\mathrm{lfm}}\frac{T_d}{2}\right] \). So the bandwidth is \(B'=K'_{\mathrm{lfm}}\cdot T_d \). It can be interpreted that the SFrFT rotates the timefrequency distribution curve of \( x\left( t\right) \) in the clockwise direction with angle \(\alpha \), so that the modulation rate and bandwidth changes.
Thus,
where \( c\left( u\right) = \int _{0}^{u} \cos \left( \frac{\pi }{2}x^2 \right) \, \mathrm{d}x\) and \( s\left( u\right) = \int _{0}^{u} \sin \left( \frac{\pi }{2}x^2 \right) \, \mathrm{d}x\) is the Fresnel integral. And
where \( u_0=f_0\sin \alpha \). So the amplitude spectrum of \({\overline{X}}_\alpha \left( u\right) \) is
and the phase spectrum is
Substituting \( K'_{\mathrm{lfm}}=\frac{B'}{T}\) into Eq. (S5):
According to the property associated with Fresnel integral, when \(B'T_d \gg 1\), i.e., \(K'_{\mathrm{lfm}}T_d^2 \gg 1\), \(c\left( u_2\right) \approx s\left( u_2\right) \approx \frac{1}{2}\), so
Therefore, when \(\left( K_{\mathrm{lfm}}+\cot \alpha /2 \pi \right) T_d^2\gg 1\), \({{\overline{X}}}_\alpha \left( u\right) \) is approximately a rectangular spectrum.
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Li, X., Wang, H. & Luo, H. A Compressed Sampling Receiver Based on Modulated Wideband Converter and a Parameter Estimation Algorithm for Fractional Bandlimited LFM Signals. Circuits Syst Signal Process 40, 918–957 (2021). https://doi.org/10.1007/s00034020015076
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Keywords
 Modulated wideband converter
 Fractional Fourier transform
 Linear frequency modulation
 Parameter estimation
 Orthogonal matching pursuit
 Cramer–Rao lower bound